Models for Social, Ecological, and Biological Systems: Narrowing the Gap Between Theory and Applications

这项研究的目的是发展数学工具，以理解和预测各种重要的和明显不同的生物现象和社会学过程的行为，如疾病的传播，生态变化，或犯罪的分布和预防。这些不同现象的一些显著特征可以用基于反应-平流-扩散方程的数学模型来描述，这些方程已被广泛用于研究生物学几个领域的基本和普遍现象，最近在社会科学中也得到了应用。虽然这些模型是现实的简化版本，但它们的数学分析有助于理解许多重要现象。与此同时，从数学的角度来看，潜在的方程可能非常有趣和具有挑战性，因为它们的解可以表现出丰富的行为，例如，模式形成和行波结构。解的行为反映了数学模型系统的关键特征，并且与现实世界系统的进化有清晰和令人信服的相似之处。然而，现实世界的系统与其更易于处理的数学模型之间仍然存在很大的差距。该项目的目标之一是通过使用积累的具体数据以及相应的数学模型校准和修改来弥合这一差距。特别重要的是对包括异质环境在内的系统的分析，对非局部分散对解决方案行为的影响的理解，以及用实际数据验证这些模型。努力将集中在四种反应-平流-扩散系统上，其中异质性是由于：生态背景下的气候变化，骚乱背景下的非局部和不对称信息传播，社会隔离背景下的收入异质性，以及由于特定集中投入造成的环境异质性。该项目的主要目标是了解异质环境和非局部扩散如何影响一般非线性反应扩散方程以及特定生态和社会背景下的解决方案模式。在整个项目中，研究人员将保持和促进与社会科学家的联系，以便在数学理论和应用之间进行急需的对话。本项目将重点发展异质环境中反应-平流-扩散系统的广泛理论，并将其应用于生态学和社会学。特别地，本研究的目的有三个方面：扩展异质环境中局部和非局部反应-平流-扩散系统的当前数学理论，通过使用这些类型的系统建模来深入了解各种（生态，社会学和生物学）复杂系统，并采取初步步骤，弥合基本数学模型与他们旨在通过结合使用数据来描述的复杂现实世界系统之间的差距。从定性的角度来看，这项工作将主要关注探索各种扩散机制对异质环境中解的传播（和缺乏）的影响：例如，确定传播速度，行波解的存在，脉动锋，行脉冲或广义锋在异质系统中的存在。此外，还对这些系统解的全局适定性、中长期渐近性和非平凡稳态解的存在唯一性等基本问题进行了严格的分析。

This research is aimed at development of mathematical tools for understanding and predicting behavior of a variety of important and apparently dissimilar biological phenomena and sociological processes, such as the spread of diseases, ecological changes, or the distribution and prevention of crime. Some salient features of these diverse phenomena can be described by mathematical models based on reaction-advection-diffusion equations that have been used extensively to investigate fundamental and ubiquitous phenomena in several areas of biology, and more recently in the social sciences. While these models are simplified versions of reality, their mathematical analysis has contributed to the understanding of many important phenomena. At the same time, the underlying equations can be extremely interesting and challenging from the mathematical point of view as their solutions can exhibit rich behaviors, e.g., pattern formation and traveling wave structures. The behavior of solutions reflects critical features of the system that mathematics models and have clear and convincing parallels with the evolution of real-world systems. However, there is still a large gap between real world systems and their more tractable mathematical models. One of the goals of this project is bridging this gap through the use of accumulated concrete data and the corresponding calibration and modification of mathematical models. Of particular importance is the analysis of systems which include heterogeneous environments, the understanding of the effects that non-local dispersal has on the behavior of the solutions, and the validation of these models with real-world data. Efforts will be focused on four reaction-advection-diffusion systems where the heterogeneities are due to: climate change in an ecological context, non-local and asymmetrical spread of information in the context of riots, income heterogeneities in the context of social segregation, and environment heterogeneities due to specific concentrated inputs. The primary goal of the project is to understand how heterogeneous environments and non-local dispersal impact solution patterns in both a general class of nonlinear reaction-diffusion equations as well in specific ecological and social contexts. Throughout this project the investigators will maintain and foster contacts with social scientists in order to conduct the much needed discourse between the mathematical theory and the applications. This project will focus on the development of an extensive theory for reaction-advection-diffusion systems in heterogeneous environments with applications in ecology and sociology. In particular, the objective of this research is three-fold: to expand the current mathematical theory for local and non-local reaction-advection-diffusion systems in heterogeneous environments, to gain insight into various (ecological, sociological, and biological) complex systems by modeling them using these types of systems, and to take an initial step toward bridging the gap between basic mathematical models and the complex real-world systems they aim to describe by incorporating the use of data. From the qualitative perspective, this work will be mainly concerned with exploring the effects that various dispersal mechanism have on the propagation (and lack thereof) of a solution in a heterogeneous environment: for example, determining the spreading speed, the existence of traveling wave solutions, pulsating fronts, traveling pulses, or generalized fronts in heterogeneous systems. Additionally, the fundamental issues of the global well-posedness of solutions to these systems, intermediate and long-term asymptotics, and existence and uniqueness of non-trivial steady-state solutions will be rigorously analyzed.